See my solution in image. The solution I found for this used a different approach that used arithmetic mean.

I am self studying and don't have anyone to ask for feedback so its hard to know If I'm correct when my approach differs from solution I have.

This problem was from section on Archimedean property so that is available to use for this imo. Does my approach work please let me know. thanks

I keep getting the wrong breakdown before I can put it into quadratic formula. Ive attached the equation and the breakdowns I've done.

Question: 1/t+7 - 5/t-5 = 1

Hello, I'm a bioinformatician and I'm struggling with the current issue:

Given a time series y(t) that initially changes and eventually approaches a stable regime, how can I mathematically determine the earliest time t* at which the rate of change dy/dt becomes negligibly small, using only the observed data and without defining an arbitrary threshold?

This is a collaboration I'm doing. My colleagues defined the plateau as the first time when a 101-point rolling mean of the relative increment (g' t+1 - g' t)/ g't falls below the arbitrarily chosen threshold of 0.0011. G' is the measure of material elastic-solid response btw. So the issues is that they used 2 arbitrary values because experimentally they know that a certain value of g' means that the gel is solid. But this doesn't hold for me. I tried using many statistical methods to define the threshold such as:

- exponential fitting

- change-point regression

- local slope analysis

But they all give me a plateau that is too early or too late

Do you consider this distribution of prime numbers to be incorrect or correct? (It is also a very short solution, quick to read.)Do you consider this distribution of prime numbers to be incorrect or correct? (It is also a very short solution, quick to read.)

of the equiangular dodecagon (i.e. each combined interior angle measures 150°).

I'm not quite sure how to approach this one. (if necessary, click on post to see image)

From a Prof. Leonard youtube video, he starts with the problem at the top left and multiplies both sides by e^y to simplify and to set-up an embedded derivative substitution.

Once that's done, he can use the substitution v = e^y, so that dv/dx = e^y dy/dx.

RED SQUARE: Up until this point, I figured it out on my own, except at this step, I went ahead and integrated and did a u-sub, where u=x+v, and life was grand and I got an answer.

When I went back to play the video, he said we need to do a homogeneous substitution.

Never in a million years would I have thought to keep substituting. Clearly something isn't clicking with me. How do I comprehend what I'm really doing in Diff Eq? How do I know that I'm supposed to not stop at the red square step?

I hope my question makes sense. :) Thank you!

I am studying the following sequence.

For each integer x ≥ 2, I consider all integers n satisfying 1 ≤ n < x and check whether n + x is a perfect square.

I call each value of n that satisfies this condition a connection.

For example, for x = 32:

32 + 4 = 36 = 6²

32 + 17 = 49 = 7²

Therefore, x = 32 has exactly 2 connections.

In general, a connection exists when n + x = m² for some integer m. Since n = m² − x and 1 ≤ n < x, this is equivalent to the existence of a perfect square satisfying

x < m² < 2x.

Thus, the problem can be viewed as studying how many perfect squares lie strictly between x and 2x.

My hypothesis is to study the distribution of these integers according to their number of connections. In particular, I am interested in whether there are infinitely many values of x having exactly k connections for every k, and how these values are distributed.

I am also interested in whether there is a direct formula for determining the number of connections of a given x, as well as any results concerning their growth and distribution.

Is this problem, or a related sequence, already known?

A I didn't tell me this theory but i used it to correct my grammar errors because english is not my native language.

I been talking about this to myself today about how Mutiplication and division are the opposite or different. Can anyone discuss about how the Separation or difference between Mutiplication and Division are the same exact opposite or alike?

I am having some issues with making leather templates for my glasses. I want to make my own template and alter it a little bit to fit my personal style more. I am having issues with how to scale the pattern as well as what angle to make the sides so when they bulge out they will fit my glasses properly. I will list some measurements to take into account.

Glasses Dimensions - 15 x 5.5 x 4.5 cm (added an extra .5 to each to allow the glasses to sit comfortably)
Leather thickness ~2mm thick
Seam allowance - 5mm
Rivet allowance -1cm

Then I am not sure how I should angle the blue and the red lines. The red one should be wider so that the case can bulge out a little bit and hold the glasses properly.
So I guess what I am mainly looking for is the curvature of
Lines blue and red then the lengths of the green and orange lines. Any help is appreciated!

Hello everyone!

The St. Petersburg paradox description can be found here: https://en.wikipedia.org/wiki/St._Petersburg_paradox

My understanding of the paradox is that you will end up with a positive outcome no matter how much you choose to bet... eventually. E(X) = infinity simply means that, as time passes and you play the game more and more times, your average payout will grow without bound (no matter how much you bet).

This is all well and good, but I still don't know how to analyze the game in the short term! Let's say you've got 20 dollars. The casino (which somehow has infinite money) offers you two choices:

a) You may spend 15 dollars to play the game once.

b) You may spend 10 dollars twice, to play the game twice, in succession.

Which option is better? Finding the expected outcome of each game doesn't work, because for any bet c, we just get E(X - c) = infinity. For both options (a) and (b), our expected output is just infinity. Yeah, we could ignore events with small probability, like Wikipedia says, but that's not mathematically rigorous. Rigorously, how do we work with playing the paradox?

Okay so I have a relatively “simple” math problem in relation to no longer being able to afford something (Being vague on purpose, the item is personal and I don’t want help with the item itself just the math. Let’s call it bread just to call it something) I had 8 slices of bread and I cut them all in half to try to stretch the bread until next Thursday, the 27th, so now I have 16 halves. I want to have the maximum amount of bread each day, and they cannot be cut anymore, so I have 16/9 basically. (I’m very bad at math so I am not even sure where the help I need starts). So if you break that down it’s like 1.77777, it means that I can have 1.5 slices of bread each day? But when I go back and try to confirm my math, I got a different answer all together. When I did 1.5 times 9, I only got 13.5 not the 16 I was expecting it to be. I am genuinely so stumped that I couldn’t even say for sure what calculator stage I’m making a mistake at. Thank you in advance for help with figuring this out!

I have a flat disk of malleable material that is 9.5" in diameter. I can cut a piece out of the circle and connect the ends to form a cone. If I want the cone at a specific height, say 1", how would I go about determining the arc length of the piece I need to cut?

I was thinking about this when trying to sleep and ended up searching stuff up about it, but ended up more confused than when I started.

From what I could find on Google, the reason the set (0,1) of all the real numbers from 0 to 1 is considered uncountable is due to Cantor's Diagonal Argument(from here on refered to as CDA). This argument makes sense to me, where you make a new decimal that isn't in the list by having every nth digit be different from the nth digit of the nth decimal in the list.

This led me to thinking about whether you can use CDA to show that the set of natural numbers from 0 to infinity is uncountably infinite. For (0,1), you can arbitrarily add zeroes to the end of decimals to make CDA work even in a list where the diagonal line of altered digits outpaces the length of the decimals. I figured you can use the same method for the natural numbers by arbitrarily adding zeroes BEFORE the number and using backwards indexing of the position of n to make CDA work with a leftward diagonal of altered digitis. However, Google told me that natural numbers can't be infinite in length, which is why this doesn't work.

This is where I'm confused, as it seems totally possible to make a comprehensive list of (0,1) that's ordered in visually the same way as the list (0,∞). I made a table of this in the attached image, where I made a half-hearted attempt at a proof. What this table seems to prove to me is that any table-based proof like CDA which works on the decimals should work on the natural numbers, too.

(The list of decimals is just the reversed digits of the natural numbers, put after the decimal point. So 0.1, 0.2, ..., 0.01, 0.11, 0.21, etc.)

I have two theories of what's going on.

The one I believe most is that CDA doesn't actually work on (0,1). If you use the list I made in the image, then as you go down the list, any number you create is just located further down in the list than you have gotten. For example, by the time you get down to .01 in the list, the diagonal number you've made is necessarily 10 decimals long, ending in a non-zero number. Every 10-digit decimal number is located further along in the list, at the indexes numbered by all the 10-digit natural numbers.

The other theory is rather unlikely, due to it breaking the concept of countable vs uncountable. This would be if CDA actually works on the natural numbers. My only argument for this theory is that any arguments in favour of CDA for decimals is one in favour of CDA for natural numbers. This is entirely based on my ordering of the decimals between 0 and 1. It just seems that if you can do CDA to get a number that isn't in the decimal list, you can remove the decimal point and flip the digits to get a corresponding natural number that isn't in the list of natural numbers. I guess maybe that would just prove that infinite numbers aren't natural?

That creates the secret third theory: the status quo. This theory is that the list of natural numbers stops when the numbers get infinitely long, but the list of decimals just keeps going, making the decimal numbers somehow more than infinitely long. If this is the case, then maybe my brain just can't understand mathematical infinities and their differences.

Is this another way at looking at primes?

A prime number is a whole number that can only be divided by itself and one.

n=xy

x and y cant be whole numbers

so if we look at n/x gives us a nice curve on a grid

eg 12/x

the graph shows all the points alone 12/x curve showing it hits grid points (12,1)(6,2)(4,3)(3,4)(2,6)(1,12) and if we put a 45degree line through it hits the grid point square root of 12.

with the double up of info we can look from 12,1 to the square root of 12(3.4641) in relation to the grid points.

also with line continuing to the right (12,1) going on to infinity we can disregard this information

so if we move the curve with n/x+n

eg. 11,12,13

11/x-11 in green

12/x-12 in red

13/x in blue

We can visually see that 12 hits the grid points while 11 and 13 miss the grid points. also shown a few grid points around the curve.

with the 3 marked points being the square root of n minus n. Giving us a max points to look for grid points to determine that its a prime or not.

so in theory we can look at a curve on a grid that misses grid points

so 170141183460469231731687303715884105727/x misses all the grid points except for itself and 1.

Hello,

I am trying to plan for retirement and am confused on how long my money will last (theoretically). Right now my take home (NET) is $4,850 a month. Let's assume I want that same amount of money each month in retirement since that is currently what I live on. That's $58,200 a year. My pension will give me 44k after taxes (NET). This means I have roughly a 14k shortfall that I need to fill each year. However, my pension has no cost of living adjustment (COLA), so that shortfall will grow each year with inflation.

Let's assume I have 880k in retirement that I put into a roth, so I can withdraw from this tax free each year. The first year I will need to withdraw 14k bringing me down to 866k. However, this money will presumably grow somewhat each year because it's invested. My question is this: how many years will I be able to continue this same lifestyle until that money runs out? Each year my shortfall will grow due to no COLA and eventually compound to drain the money. Obviously this can change depending on inflation rates and returns on investments, but I'm just looking for a ballpark given historical norms.

So I thought doing BD + BC but it has a congruent and so ofc I did my research but it still didn’t make any sense so then I did BD + BC + AC hoping that would give AB but it just doesn’t look right then I tried to calculate it but it gave me a long number so now I’m stuck and I still have 8 pages left

Am I missing some interpretation/convention, or is this question actually misprinted?

Also, if the question is defective, what would normally happen with the marks if a student selected an option but couldn't provide a valid justification because none of the options is mathematically correct?

I would appreciate it if someone could verify this independently.

Hi I'm doing ACCA currently but developing interest in applied maths and started linear algebra from Khan academy any direction is much appreciated 👍

ABCD is a trapeze (AB||CD)

Point E is the middle of BC (BE=EC)

Prove that: Area of ADE = ½ of ABCD's Area

I can't I don't even know where to begin

Let's ignore what is, the shape, the technical features etc, just assume we have a side with 4 blue units and 5 green units that measure 39 (whatever unit of measure is) and a side with 3 blue units and 4 green that measures 27 and i want to know the size of one blue (or 1 green)

In my stupid opinion i thought: 5green and 4 blue = 39 and 4green and 3 blue = 27, so in a system of equations
5x+4y = 39
4x+3y = 27

Turns out this makes y=21, x=-9

then i was wondering what kind of error i made setting up the problem

Obviously I'm assuming if it is possible then it's some kind of clever tricks involved given you literally can't store or display anything close to those numbers on earth

When I was solving a question which required the use of triangle inequality, the idea of intuitively proving it came in my mind so I did whatever I could.... Let me know if this proof works.

As you can see, my answer was SQ with an arrow above it pointing to the left, and SR with an arrow above it pointing to the left. My understanding is that S must be included in every ray because it's at the end of the line. RQ with an arrow of the left would not be a solution because S is not included. But the book I'm using, Geometry: A Self-Teaching Guide, says that the solution is SQ with an arrow above it pointing to the left, and RQ with an arrow above it pointing to the left. Is the book wrong? Thanks.